1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6.

7

Form the alternating sum of blocks of three from right to left.

1,369,851: 851 − 369 + 1 = 483 = 7 × 69

Subtract 2 times the last digit from the rest. (Works because 21 is divisible by 7.)

483: 48 − (3 × 2) = 42 = 7 × 6.

Or, add 5 times the last digit to the rest. (Works because 49 is divisible by 7.)

483: 48 + (3 × 5) = 63 = 7 × 9.

Or, add 3 times the first digit to the next. (This works because 10a + b − 7a = 3a + b − last number has the same remainder)

483: 4×3 + 8 = 20 remainder 6, 6×3 + 3 = 21.

Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, -1, -3, -2 (repeating for digits beyond the hundred-thousands place). Then sum the results.